Committee Chair

Cox, Christopher

Committee Member

Cetinkaya, F. Ayça; Mukherjee, Rick; Liang, Yu

Department

Dept. of Computational Science

College

College of Arts and Sciences

Publisher

University of Tennessee at Chattanooga

Place of Publication

Chattanooga (Tenn.)

Abstract

Inverse quantum problems ask whether a central potential V(r) can be reconstructed from limited spectral information under explicit modeling assumptions. This dissertation develops and validates a reproducible, transform-based framework for reconstructing ground-state radial densities rho(r) = |R(r)|^2 and central potentials V(r) from discrete bound-state energy levels using generalized Bertlmann-Martin (GBM) moment inequalities as physics-informed constraints. In the Laplace route, L(q) is assembled from moment data and inverted numerically to recover rho(r). In the Fourier/form-factor route, even moments determine F(q) and inverse transformation yields rho(r). To support sparse spectral inputs, analytic continuation is performed with Pade approximants, and the framework is organized into staged, reproducible reconstruction modes ranging from simple baseline cases through analytic, Pade, hybrid, and GBM-constrained settings that isolate error sources due to numerical inversion, Pade approximation, GBM even-moment estimation, and odd-moment completion strategies. A cross-potential study covers seven potentials: five main benchmark potentials (Coulomb, harmonic oscillator, Kratzer, Hulthen, and the canonical hyperbolic molecular well) plus supplementary hyperbolic variant and intermediate cases. Within that scope, the dissertation uses common baselines and transform-aware diagnostics to identify where continuation, inversion, admissibility, and moment-completion choices limit reconstruction quality under the reported settings. Across the five main benchmark potentials, the Laplace (GBM) route yields lower relative-L2 V(r) error than the Rohrl-style LSQ comparator under the shared settings, while Laplace and Fourier comparisons remain close with potential-dependent winners. The work contributes (1) a consistent radial-orbital notation framework that reconciles inverse-theory and physics conventions, (2) a reproducible framework for auditing each error source, and (3) practical guidance on when Pade approximation dominates error versus when inversion stability, moment estimation, or tail handling are limiting factors under the stated benchmark conditions. In addition, a limited core-Coulomb minimum-level comparison is reported: under fixed Laplace GBM and Fourier GBM configurations, k = 6 is the first passing level-count threshold while k = 4 fails in both transforms. Broader minimum-level and excited-state comparisons remain supplementary extension cases rather than established benchmark results.

Acknowledgments

I thank my co-advisors, F. Ayça Çetinkaya and Chris Cox, for their guidance and support. I thank my committee members, Rick Mukherjee and Yu Liang, for their time and feedback. I also acknowledge institutional support from the UTC Quantum Center and the UTC Department of Mathematics. No external funding supported this dissertation.

Degree

Ph. D.; A dissertation submitted to the faculty of the University of Tennessee at Chattanooga in partial fulfillment of the requirements of the degree of Doctor of Philosophy.

Date

5-2026

Subject

Fourier transformations; Matrix inversion; Sparse matrices; Reconstruction (Graph theory); Transformations (Mathematics)

Keyword

quantum inverse problems; quantum potential reconstruction; sparse spectral data; generalized Bertlmann-Martin moment inequalities; inverse Laplace transform; inverse Fourier transform

Document Type

Doctoral dissertations

DCMI Type

Text

Extent

xv, 92 leaves

Language

English

Rights

http://rightsstatements.org/vocab/InC/1.0/

License

http://creativecommons.org/licenses/by/4.0/

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